Learn Danish: Jeg kan cykle i parken. - I can cycle in the park.

Because kan is a modal verb. Just like in English where we say 'I can cycle' instead of 'I can to cycle', Danish drops the word at (to) after modal verbs. You just use the bare infinitive form: cykle.

Does cykle just mean 'cycle'? What if I want to say 'ride a bike'?

You still use cykle! Danish uses this single verb for both 'to cycle' and 'to ride a bike'. You shouldn't try to translate 'ride a bike' word-for-word, because the Danish verb at ride is only used for riding horses or other animals.

Why does the sentence use parken instead of a separate word for 'the'?

In Danish, the definite article ('the') is usually attached to the end of the noun. Since 'a park' is an en-word (en park), you take that en and stick it on the end to get parken (the park).

Where would I put ikke if I wanted to say 'I cannot cycle in the park'?

You place ikke (not) directly after the first verb, which is the modal verb kan. The negative sentence would be: Jeg kan ikke cykle i parken.